LOCAL NILPOTENCY IN VARIETIES OF GROUPS WITH OPERATORS

A theorem of a rather general nature is proved, which gives a positive solution to the restricted Burnside problem for a variety of groups w ith operators whose identities are obtained by ''operator diluting'' ( in some precise sense) ordinary identities defining a variety of group s for which this problem has a positive solution. Namely, let OMEGA be a finite group, V a family of OMEGA-operator identities, and VBAR a f amily of (ordinary) group identities obtained from V by replacing all operators by 1. Suppose that the associated Lie ring of a free group i n the variety MBAR defined by VBAR satisfies a system of multilinear i dentities that defines a locally nilpotent variety of Lie rings with a function f(d) bounding the nilpotency class of a d-generator Lie ring in this variety. It is proved that if, for a d-generator OMEGA-group G, the semidirect product G lambda OMEGA is nilpotent, then the nilpot ency class of G is at most f(d . (Absolute value of OMEGA(Absolute val ue of OMEGA - 1)/(Absolute value of OMEGA - 1)). A strong condition th at G lambda OMEGA be nilpotent is automatically satisfied if both G an d OMEGA are finite p-groups. Instead of the condition on the identitie s of the associated Lie ring, an analogous condition on the identities VBAR could be required, but such a condition would be stronger. An ex ample at the end of the paper shows that the word multilinear in this condition is essential. It is not yet clear whether the condition that OMEGA be finite is essential, and whether one can choose a function f rom the conclusion to be independent of Absolute value of OMEGA. Earli er, in [1], a similar theorem on nilpotency in varieties of groups wit h operators was proved by the author. The author's results on groups w ith splitting automorphisms of prime order p (see [2], [3]) are protot ypes for both papers on operator groups. Bibliography: 18 titles.